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Polyhedra and packings from hyperbolic honeycombsMartin Cramer Pedersena,1 and Stephen T. Hydea

aDepartment of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 25, 2018 (received for review November 29, 2017)

We derive more than 80 embeddings of 2D hyperbolic honey-combs in Euclidean 3 space, forming 3-periodic infinite polyhedrawith cubic symmetry. All embeddings are “minimally frustrated,”formed by removing just enough isometries of the (regular,but unphysical) 2D hyperbolic honeycombs {3, 7}, {3, 8}, {3, 9},{3, 10}, and {3, 12} to allow embeddings in Euclidean 3 space.Nearly all of these triangulated “simplicial polyhedra” have sym-metrically identical vertices, and most are chiral. The most sym-metric examples include 10 infinite “deltahedra,” with equilateraltriangular faces, 6 of which were previously unknown and someof which can be described as packings of Platonic deltahedra. Wedescribe also related cubic crystalline packings of equal hyper-bolic discs in 3 space that are frustrated analogues of optimallydense hyperbolic disc packings. The 10-coordinated packings arethe least “loosened” Euclidean embeddings, although frustra-tion swells all of the hyperbolic disc packings to give less densearrays than the flat penny-packing even though their unfrustratedanalogues in H2 are denser.

hyperbolic geometry | nets | minimal surfaces | graph embeddings |symmetry groups

Triangulations are central constructions in diverse areas ofpure and applied sciences, from cartography (1) and signal

processing (2) to fundamental mathematics (3). Triangular poly-hedra characterize many spatial packings, such as the icosahedralarrangement of discs on a sphere and the penny packing in theflat plane and close packing of equivalent spheres in Euclidean3 space (4), E3. The Platonic triangular polyhedra can be assem-bled into numerous configurations, many relevant to geometriesof crystalline, quasi-crystalline, and disordered packings (5–8).Consequently, triangulated structures are common in condensedatomic and (bio)molecular assemblies. Triangular polyhedra arefound in glasses and random dense sphere packings (9–12). Theyare essential building blocks in tetrahedrally close-packed struc-tures in alloys and soft materials (13–15); Goldberg polyhedra(16), which describe the structures of many viruses (17, 18);and Boerdijk–Coxeter helices (19) in biological fibers (20) andnanowires (21).

Here, we derive a large number of infinite, crystalline pat-terns, namely nets, triangular infinite polyhedra, and associateddisc packings in 3D Euclidean space, E3, derived from Coxeter’s“regular honeycombs” of 2D hyperbolic space (22), H2. These 3-periodic crystalline structures minimize the geometric frustrationthat results from mapping H2 to E3. Thus, they are importantadditions to the compendium of regular patterns.

The construction is done in two stages. First, we revisitdense disc packings of 2D hyperbolic space, first explored byand Coxeter (22) and Toth (23). We deform related hyper-bolic nets whose edges link adjacent discs to realize the netsand their associated disc packings in E3. Finally, we relaxsymmetrized versions of the nets in E3, to recover as simi-lar edge lengths as possible. In those cases where equal edgesresult, the nets define edges of infinite triangular polyhedra,“deltahedra.” More commonly, frustration imposes unequaledges, describing the skeleta of infinite simplicial polyhedra.An extraordinary wealth of disc packings, nets, and polyhe-dra emerge from just a small number of hyperbolic triangularhoneycombs. The presented methods are readily extended torealize patterns in E3 from patterns in H2, beyond triangularhoneycombs.

Disc Packings and Triangular PatternsThe density of 2D hard disc packing is characterized by the ratioof the total area of the packed objects to the area of the embed-ding space. Thus, the hexagonal “penny packing” of equal discsrealizes the maximal packing density in the plane, E2 (24, 25).Dense packings of equal discs on the surface of the 2 sphere,S2, are more subtle, since the sphere’s finite area means thatoptimal solutions depend on the disc radius. The formal defini-tion of packing density for equal discs in the third homogeneous2D space—the hyperbolic plane, H2—is also complicated by thenature of that space. Nevertheless, Toth (23) established thatpacking densities in all three homogeneous 2D spaces can beunified via a simple formula, which gives an upper bound for thedensity of any packing in S2, E2, or H2,

ρ (N )=3 csc

(πN

)− 6

N − 6, [1]

where N > 3 denotes the coordination number of the disc pack-ing, equal to 6 for the penny packing, whose density is ρ(6)=π√12≈ 0.91. We can associate a “regular” skeletal net with the

penny packing, whose vertices align with the penny centers andwhose edges join pennies in mutual tangency. The net has thetopology and geometry of the edges of the equilateral triangulartiling of the Euclidean plane, E2, denoted 36. Here, “regularity”implies symmetrically identical faces, edges, and vertices. The 36

triangulation can be realized by repeated reflections in the edgesof a triangular kaleidoscope with vertex angles π

2, π3

, and π6

: Theregular pattern has orbifold ∗236 (6).

Regular triangulations 3N and their regular skeletal nets(denoted {3,N }) are realizable for all integer values of N ≥ 3(26). They are all kaleidoscopic patterns—characterized by tri-angular kaleidoscopic orbifolds ∗23N, illustrated in Fig. 1. Forexample, the regular triangulations 3N , N =3, 4, 5 consist ofspherical equilateral triangular faces (in S2). If we scale S2 so that

Significance

The simplest 2D regular honeycombs are familiar patterns,found in an extraordinary range of natural and designedsystems. They include tessellations of the plane by squares,hexagons, and equilateral triangles. Regular triangular hon-eycombs also form on the sphere; they are the triangularPlatonic polyhedra: the tetrahedron, octahedron, and icosa-hedron. Regular hyperbolic honeycombs adopt an infinitevariety of topologies; these must be distorted to be sit-uated in 3D space and are thus frustrated. We constructminimally frustrated realizations of the simplest hyperbolichoneycombs.

Author contributions: M.C.P. and S.T.H. designed research, performed research, analyzeddata, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 To whom correspondence should be addressed. Email: martin.pedersen@anu.edu.au.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1720307115/-/DCSupplemental.

Published online June 20, 2018.

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Fig. 1. (A) Regular triangulations 3N formed by decorating an orbifold∗23N by an edge and vertex. (B and C) The 35 triangulation of S2 (B) and36 tiling of E2 (C). (D) The analogous triangulation 37 of H2 (drawn in thePoincare disc model).

the triangles’ edges are of unit length, related dense disc pack-ings are realized by locating unit discs on the sphere whosecenters coincide with vertices of the triangulation. These pack-ings have densities precisely equal to ρ(N ) in Eq. 1. So regulararrangements of equal discs realize the densest packings in bothS2 and E2. Similarly, the densest disc packings for N > 6 areregular packings in H2, and their densities are equal to ρ (N ).Their skeletal nets define regular triangulations of H2 for allintegers N exceeding 6. In summary, densest 2D packings arerealized by centering equal discs on the vertices of regular tilingsthat describe triangular honeycombs, {3,N }, regardless of their2D geometry (S2, E2, or H2). Note that since all patterns 3N

with N > 6 inhabit H2, the overwhelming majority of regularhoneycombs are hyperbolic.

Regular hyperbolic honeycombs have been explored consid-erably less than their Euclidean relatives, in part due to the“unphysical” nature of H2. H2 cannot be smoothly embeddedin E3, in contrast to S2 and E2, both of which can be embed-ded in E3 isometrically. (The exact result, due to Hilbert andlater refined by Efimov, is discussed in ref. 27.) Just as anymap of the globe (S2) onto a flat sheet of paper (E2) is sub-ject to distortion, H2 cannot be mapped into E3 without some“frustration.” Geographers have found ways around the formerproblem, and here we describe a number of embeddings of H2

into E3 that minimize, but cannot completely remove, the frus-tration imposed by the embedding from H2 into E3. We exploreregular patterns for N =7, 8, 9, 10, 12 whose distortions are justsufficient to embed them in E3, and no more. These “minimallyfrustrated” structures include a number of symmetric hyper-bolic triangular polyhedra 3N , their associated nets [3,N ], andrelated disc and sphere packings in E3. We denote the result-ing Euclidean nets [3,N ], in contrast to their regular hyperbolicantecedents, {3,N }.

Mapping from H2 to E3: Commensurate SubgroupsImagine a pattern in H2, analogous to symmetric patterns inthe plane or on the sphere. Since all embeddings of H2 intoE3 are frustrated, the pattern is necessarily distorted in mov-ing from H2 to E3. However, some hyperbolic patterns may bemoved into E3 without losing any of their isometries, providedthe frustration itself adopts those isometries. Hyperbolic pat-terns are “commensurate” if they can be embedded in E3 suchthat all of their 2D (in-surface) isometries are retained in E3.The 2D isometries of generic patterns, whether in S2, E2, orH2, are encoded by orbifolds (6). For example, the regular 3N

triangulations of the sphere (N =3, 4, 5) are all commensurate,since patterned spheres with tetrahedral, octahedral, and icosa-hedral point groups have orbifolds ∗23N (28). We can orderpatterns from most to least symmetric by the orbifold charac-teristic of their associated orbifold, χorb , readily calculated fromthe Conway symbol of the orbifold (6). For example, the moresymmetric the spherical pattern the smaller the area of an asym-metric domain, equal to 2πχorb , normalized to spheres of unitcurvature. Likewise, we can order patterns in H2 by decreasingsymmetry (28). In contrast to S2, orbifolds in H2 have |χorb | ≥ 1

84,

where the lower bound is realized by the most symmetric hyper-bolic pattern, ∗237. Coincidentally, this is the orbifold for thesimplest regular hyperbolic honeycomb in H2, {3, 7}. Any 2Dhyperbolic pattern must have this or lower symmetry, in the senseintroduced above. But ∗237 is incommensurate, since it is impos-sible to embed a hyperbolic surface in E3 with this orbifold.Likewise, ∗238, ∗239, ∗23(10), and ∗23(12) are incommensurate.In fact, the most symmetric commensurate hyperbolic orbifold isthe ∗246 orbifold (with |χorb |=1/24) and all other hyperbolicsurfaces embedded in E3 have |χorb |> 1/24. The pattern ∗246characterizes the 2D isometries of the most symmetric hyper-bolic surfaces in E3, the cubic triply periodic minimal surfaces(TPMS), known as the D , P , and Gyroid surfaces (29, 30). Anal-ysis of the intrinsic symmetries of these and other TPMS, as wellas 2-periodic hyperbolic “mesh” surfaces, has led to enumera-tion of the most symmetric commensurate hyperbolic orbifolds(28, 31, 32). The degrees of frustration required to render theregular hyperbolic honeycombs commensurate are dependent onthe degree of symmetry breaking required to build the pattern inE3. We ranked the frustration via the index of a commensuratesubgroup relative to its unfrustrated, regular, parent honey-comb in H2: Increasing index implies more frustration, sincemore isometries are lost. We generated increasingly frustratedpatterns by systematically dropping isometries of the regularhoneycombs, using the GAP software (33) to construct a latticeof subgroups of the parent honeycombs. The subgroups were

Fig. 2. (A–E) Dense N-coordinated disc packings in H2 (drawn in thePoincare disc model of H2) for N = 7, 8, 9, 10, and 12, together with reg-ular {3, N} net. (F–J) Embeddings of minimally frustrated disc packings inH2, whose symmetries produce embeddings with commensurate subgroupsof E3, together with related irregular [3, N] nets. (K–O) Embeddings of thefrustrated disc packings in E3, as coverings of 3-periodic minimal surfaces.

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identified via their 2D hyperbolic orbifolds. All subgroups withχ≥− 1

6were generated, since that constraint was found to be

sufficiently generous to determine the least frustrated embed-dings of hyperbolic triangular honeycombs. Among these, wefound seven distinct commensurate orbifolds, including 2, 5, 4,1, and 1 for {3,N } honeycombs with N =7, 8, 9, 10, 12, respec-tively. Those orbifolds, and their associated subgroups on TPMS,give multiple embeddings in E3. We catalog the associated discpackings, nets, and polyhedra emerging in E3 from the fiveregular hyperbolic honeycombs below.

A Worked Example for the 310 PatternThe procedure is described in detail in SI Appendix; relevantgroup–subgroup lattices and lowest index commensurate sub-groups are described in SI Appendix, Figs. S5–S9). To guidethe reader, we first briefly describe our constructions of discpackings, nets, and polyhedra derived from one of the regularhyperbolic honeycombs: the 310 hyperbolic honeycomb.

GAP detected six subgroups of the group of the regularhoneycomb [orbifold symbol ∗23(10)] whose orbifolds had char-acteristics |χorb | ≤ 1

6(listed in SI Appendix, Fig. S8 and Table

S5). Of those, only the subgroup with largest index (5, orbifoldsymbol 3 ∗ 22) was commensurate; this is the intrinsic symme-try of minimally frustrated patterns. This orbifold is unusuallysimple, as it can be cut open to give a unique pentagonal formin H2 that is a union of four ∗246 triangles. By design, thisdomain was also a union of five ∗23(10) triangles, and a suitabledecoration by (three partial) edges and a (quarter) vertex wasinferred from the decoration of edges and vertices within each∗23(10) triangle in its parent regular {3,N } net (SI Appendix,Fig. S15). The resulting frustrated, irregular, commensurate[3, 10]net in H2 was vertex transitive and edge-3 transitive, withunequal edges. Disc packings in H2 were constructed by placinghyperbolic discs at the vertices of the regular {3,N } and frus-trated irregular [3, 10] nets. The radius of congruent discs wasmaximized to form packings. The frustration induced by the com-mensurate disc packing opens fissures in the disc array, resultingin an 8-coordinated packing, with slightly enhanced symme-try compared with the underlying edge net, namely ∗344. Thisis the densest commensurate embedding of the regular dense10-coordinated regular disc packing, filling 87% of H2, comparedwith the regular dense filling fraction of 93%. We mapped thatcommensurate [3, 10]net and associated quasi-dense disc pack-ing onto the P , Gyroid , and D TPMS, guided by the underlying∗246 triangles and their placement on the TPMS (34) formingthree topologically distinct disc packings (SI Appendix, Table

S7) and curvilinear 3-periodic nets in E3, one on each TPMS.The disc packing on the P is shown in Fig. 2N. The 3-periodiccurvilinear nets had space groups induced by the combination oforbifold (3 ∗ 22) and TPMS (28): namely Pm3, P23, and I 213for the P , D , and Gyroid . Those nets were straightened in E3,conserving all isometries induced by those space groups, to yieldnets with unit average edge length and minimal differences inedge lengths (SI Appendix, Table S11). The net induced by theP embedding relaxed to give strictly equal edges; pairs of edges“collapse” to a common edge, so that nine geometrically distinctedges emerge from each vertex. The (10-valent) nets derivedfrom the D and Gyroid patterns have variable edge lengths.These net data are shown in SI Appendix, Fig. S17 and listed inSI Appendix, section S4.35. To build the three minimally frus-trated 310 polyhedra, we inserted triangles in the faces of therelaxed nets. The resulting “simplicial” polyhedra derived fromthe Gyroid and D (Fig. 3 H and I) patterns are open 310 sponges,with both irregular and regular triangular faces. Pairs of faces ofthe D polyhedron are coplanar: It can be described as a sim-ple cubic array of Archimedean cuboctahedra sharing commonsquare faces, with half of their triangular faces removed, leav-ing a pair of diamond labyrinths. The P polyhedron (Fig. 4, 9)is a 310 “deltahedron” with equilateral triangular faces, forminga simple-cubic array of edge-shared convex Platonic icosahedra.In summary, the essential frustration imposed on the regular 310

hyperbolic honeycomb to embed it in E3 results in irregular—although very symmetric—8-coordinated disc packings, degree-9and -10 nets, and infinite polyhedra with both open and closedcellular morphologies, containing simpler convex Platonic andArchimedean polyhedra. The constructions follow a similar pathfor all honeycombs. The minimally frustrated 37, 38, and 39

honeycombs are much richer than the 310 case, for three rea-sons. First, more than one maximally symmetric commensurateorbifold was detected (SI Appendix, Figs. S5–S7). Second, multi-ple groups shared some of those orbifolds, leading to multiple[3,N ] nets in H2 (SI Appendix, Fig. S11). Third, the orbifold2223 can be embedded in infinitely many ways on the TPMS,leading to a (finite) number of topologically distinct embed-dings of the nets in E3. Like 310, the 312 honeycomb gave justone least frustrated commensurate orbifold, with unique formin H2. We report these constructions derived from all triangularhoneycombs, {3,N }, where N =7, 8, 9, 10, 12.

Minimally Frustrated Hyperbolic Disc Packings in E3

Consider the unfrustrated, N -coordinated, regular, dense discpackings in H2. The skeleta of those packings form [3,N ] nets,

Fig. 3. Some cubic, vertex-transitive, infinite, simplicial polyhedra formed from symmetrized [3, N] net embeddings in E3, produced by relaxing triangulatedreticulations of the P, D, and Gyroid TPMS. Spheres are located at vertices for clarity. P, D, and (the pair of) Gyroid embeddings are red, green, blue, andyellow (as in SI Appendix, Fig. S17). (A–D) Four distinct 37 polyhedra produced from the same hyperbolic pattern on the 222301 orbifold. (E and F) 38

patterns via the 222314 orbifold and the 222321 orbifold. (G) The 39 polyhedra via the 222331 orbifold. (H and I) 310 polyhedra on the 3 ∗ 22 orbifold. (J) A312 polyhedron formed by decorating the 2*33 orbifold.

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Fig. 4. Deltahedra, color coded by their TPMS surface embeddings.

with a disc centered on each vertex. The Euler characteristicper vertex, χV =1− N

2+ N

3= 6−N

6, since the total number of

edges and faces per vertex is N2

and N3

, respectively. The sym-metries of the unfrustrated nets were broken, without changingtheir topologies [3,N ], to form a commensurate subgroup, whoseorbifold had characteristic χorb . Since each net vertex representsa (fractional) disc, the fractional number of discs per orbifold isgiven by 6χorb

6−N. Quasi-dense packings in E3 were built by locating

6χorb6−N

discs within a single commensurate orbifold whose equalradii were maximal and then embedding the orbifold, and itscopies, on the TPMS. The resulting frustrated disc packings arenecessarily less dense than their ideal counterparts in H2, sincesome symmetry breaking was required to embed them in E3. Asa result, they were no longer N coordinated.

The unfrustrated, dense, 7-coordinated disc packing in H2 isshown in Fig. 2A. This regular pattern is characterized by theorbifold ∗237; its minimally frustrated commensurate counter-parts are described by two groups with common orbifold 23×and one with orbifold 2223 (SI Appendix, Fig. S6). A frus-trated dense packing of equal hyperbolic discs, with orbifold2 ∗ 23, is commensurate with all three of these commensurategroups, since the (group associated with) orbifold 2 ∗ 23 is anindex-2 supergroup of both 23× and 2223. We tuned the discradius and location to give maximum coverage of the 2 ∗ 23 orb-ifold, consistent with the requisite disc fraction per orbifold. Theresulting quasi-dense, frustrated disc packing, whose underlying

deformed [3, 7] net has orbifold symbol 2223, has coordination4, as shown in Fig. 2B. The frustrated packings for 8- and 9-coordinated patterns adopt the commensurate orbifold, ∗246,and the 12-coordinated packing forms a minimally frustratedembedding in 3 space with orbifold ∗266. The minimally frus-trated [3,N ]nets have orbifolds 2 ∗ 23, 2 ∗ 23, and 2 ∗ 33, for N =8, 9, and 12, respectively (SI Appendix, Figs. S6, S7, and S9).(The 10-coordinated pattern is discussed in the previous section.)The structural data for all packings are listed in SI Appendix,Table S7. Details of the frustrated disc packings are listed inTable 1.

Some examples of the associated minimally frustrated discpackings are drawn in H2 in Fig. 2 F–J. Since all of the minimallyfrustrated orbifolds were either coincident with the hyperbolicsymmetries of the D , Gyroid , and P cubic TPMS (∗246) or sub-groups thereof, they can be mapped onto any one of those cubicTPMS, giving three alternative embeddings in E3 of the regularpackings for each coordination number, N . A selection of thosepatterns is illustrated in Fig. 2.

Minimally Frustrated [3, N]NetsThe deformed [3,N ] nets induced by these commensurate discpackings lie in the TPMS, and their net geometry was fixed bytheir orbifold symmetry and had maximal 2D density. We relaxedthose nets in E3 to build embeddings that both were maximallysymmetric in E3 and had—as far as possible—equal edge lengths.

First, the frustrated [3,N ] nets were constructed in H2 withcommensurate symmetries. Fundamental domains of the netswere inferred from decorations of the unfrustrated parent ∗23Norbifolds. That process, illustrated in SI Appendix, Fig. S3 C andD, led to one or more motifs for all hyperbolic crystallographicorbifolds (SI Appendix, Figs. S10–S16). The extended [3,N ]netsin H2 are the orbits of the relevant decoration by the actionof the groups associated with these crystallographic hyperbolicorbifolds. This process is illustrated in Fig. 5 A and B. Second,those hyperbolic nets were mapped to E3, via the TPMS, as fol-lows. The 3-periodic P , D , and Gyroid surfaces are covers ofa common genus-3 tritorus, built of 96 ∗246 domains. The tri-torus is built from a hyperbolic dodecagon in H2, compactified(or “glued”) via six hyperbolic translations (36). The edges of theinfinite hyperbolic net contained within the dodecagon thereforealso glue up in pairs to form a finite quotient net, modulo thosesix hyperbolic translations corresponding to the gluing vectors.The hyperbolic quotient graphs can be embedded as a reticula-tion of this tritorus, and we label edges traversing the boundariesof the dodecagon by their associated hyperbolic translation. Wedetermine those labeled quotient graphs from the net fragment(plus dangling edges) contained within this tritorus, as sketchedin Fig. 5C. At this stage of our constructions, each parent [3,N ]net was mapped into a 3-periodic net in E3, whose topology wasdetermined by the arrangement of the hyperbolic dodecagon rel-ative to the underlying commensurate hyperbolic [3,N ]net, andthe surface map. The final step was to embed the nets of fixedtopology into E3 and build triangulations and packings fromthose skeleta. The 3-periodic nets in E3 were constructed bymapping the hyperbolic translations to lattice vectors in E3,

Table 1. Dense ideal N0-coordinated hyperbolic disc packings inH2 and their minimally frustrated N-connected embeddings in E3

N0 Fig. ρ0 N Fig. ρ ρ/ρ0

7 2A 0.914 4 2F 0.774 0.8468 2B 0.920 4 2G 0.674 0.7339 2C 0.924 6 2H 0.828 0.89710 2D 0.927 8 2I 0.872 0.94012 2E 0.932 6 2J 0.732 0.796

The regular and frustrated 2D packing densities are denoted ρ0 and ρ,respectively (where ρ0 is given by Eq. 1).

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Fig. 5. Construction of minimally frustrated [3, N] nets from a commen-surate subgroup. (A) A ∗2224 fundamental domain in H2 shown in gray,decorated with a motif (in red) derived from subgroup 9 in SI Appendix,Table S3. (B) The orbit of the decorated orbifold is an embedding of the [3, 8]

graph in H2. (C) A dodecagonal domain that builds a single genus-3 unit cellof the D, Gyroid, and P TPMS. The graph edges within that domain of H2

define the quotient graphs of infinite 3-periodic nets in H2, modulo the sixtranslation vectors joining opposite edges of the dodecagon (SI Appendix,Fig. S2). (D) The Euclidean quotient graphs in C were symmetrized using theSystre algorithm (35), giving embeddings in E3. In this example, we showthe embedding via the P surface.

described in ref. 34 (listed in SI Appendix, Table S1). The map-ping is dependent on the TPMS, and a pair of maps are possiblein general on the Gyroid (37). This step therefore produces threeor four distinct labeled quotient graphs. Those labeled graphswere input into the Systre package, which relaxes 3-periodicnets in E3 to give explicit embeddings with maximal crystallo-graphic symmetry (38). We imposed a heavy cost on unequaledge lengths during relaxation to produce, where possible, netembeddings with approximately equal edges. The example in Fig.5C, realized on the P TPMS, was processed by Systre to producea cubic net, of known topology, listed in Reticular ChemistryStructure Resource (RCSR) as dgp (39). In general, a hyper-bolic orbifold can be embedded in more than one way into theTPMS, so our construction is not unique. (This complicationdoes not arise for the construction of the disc packings above,since their relevant orbifolds have unique embeddings.) Amongthe orbifolds identified here, the 2223 orbifold admits multi-ple embeddings, on the P , D , and Gyroid (40). The geometricscheme is outlined in SI Appendix, Fig. S4. Due to computationalconstraints we have constructed a restricted suite of patterns,indexed as 01, 11, 21, 31, and 41 for N =7, 8, 9 as well as 12,13, 14, 23, and 32 for N =7 and N =8, using the notationof ref. 40. The source of potential multiplicity from nontriv-ial automorphisms of the embedded subgroups in the surfacesdescribed in ref. 34 does not lead to any distinct patterns forour subgroups.

We built embeddings from 36 minimally frustrated hyper-bolic quotient graphs: 12, 14, 8, 1, and 1 graph(s) for N =7, 8, 9, 10, 12, respectively. The final embeddings generated bySystre—optimized to minimize the variations of edge lengths—retained their maximal symmetry consistent with their topology.We quantified the homogeneity of edge lengths by the ratio ofthe shortest edges to the average edge length. Most of these poly-hedra have unequal edge lengths. In the spirit of nets reported inRCSR (39), we distinguished “good” skeletal nets from others:The shortest vector between polyhedral vertices in a good net isspanned by an edge. These data are summarized graphically inSI Appendix, Fig. S17. The nets are 3 periodic, with cubic symme-try, containing 24, 12, 8, 6, and 4 vertices per unit cell for N =7,8, 9, 10, and 12, respectively. In some cases, the Euclidean netsform multigraphs, with double edges between the vertices. Theresulting Systre embeddings in E3 collapse those edges to a singlecommon edge, giving nets of reduced degree compared with theirparent patterns. The resulting nets display a range of geometriesand symmetries, both chiral and achiral. In some cases, identicalnets formed in E3 from hyperbolic nets with different commensu-rate subgroups: We constructed 136 nets that relaxed to give 86distinct examples. As they derive from honeycombs, the parenthyperbolic nets are regular. With the exception of 7 nets (withtwo distinct vertices), their minimally frustrated embeddings inE3 are vertex-1 transitive, yet their edge transitivities vary from

2 to 8. Detailed data for the nets can be found in SI Appendix,Tables S8–S12.

Vertex-Transitive Infinite 3N PolyhedraThe 3 cycles of these minimally frustrated [3,N ] nets can befaceted, giving infinite triangulated polyhedra which in manycases are embedded in E3. These polyhedra are reminiscent ofCoxeter’s “skew polyhedra” (41), since renamed “skew apeiro-hedra.” Other examples of semiregular infinite polyhedra havebeen found (42). Since our infinite polyhedra are frustrated,many examples—although not all—contain triangular facetswith unequal sides. We call such polyhedra simplicial polyhe-dra, while the cases whose faces are equilateral are nameddeltahedra (43). The study of deltahedra has a long history.The Platonic tetrahedron, octahedron, and icosahedron areamong the eight convex deltahedra (44). An infinite number ofnonconvex deltahedra can be constructed, including the cele-brated stella octangula (45) and the Boerdijk–Coxeter helices(19). Apart from the tetrahedron, octahedron, and icosahe-dron, just one further vertex-transitive deltahedron is usuallylisted: the great icosahedron, one of the four regular Kepler–Poinsot stellated (and nonconvex) polyhedra (46). Higher genus“toroidal” deltahedra are known, although scarce: An exam-ple can be constructed from eight regular octahedra (47). The[3,N ] nets we have constructed afford a wealth of hyper-bolic equilateral-triangulated infinite polyhedra analogous tofinite deltahedra. Since these infinite 3N polyhedra inherit thesymmetries of their skeletal nets, they are largely uninodal, asare the regular triangular polyhedra and the great icosahedron.These infinite triangular polyhedra conserve the [3,N ] topologyof their parent honeycombs, in contrast to the altered topologiesof the frustrated disc packs and edge-collapsed nets. Since theiredges are not symmetrically equivalent, they are irregular.

A selection of infinite simplicial polyhedra are shown in Fig.3. Many of them resemble infinite cubic arrays of finite convexpolyhedra; these examples are, however, single (infinite) poly-hedra. In some cases (indicated in SI Appendix, Fig. S17), theSystre relaxation gives overlapping edges and vertices; e.g., theP embedding of the 312 polyhedron (10 in Fig. 4) containsadjacent triangular faces folded over their common edge by adihedral angle of π, so that the faces coincide. Infinite polyhe-dra derived from such collapsed edges, like this 312 polyhedron,are strictly not embedded in E3, due to their self-intersections;they are “immersed.” These include packings of irregular tetra-hedra and octahedra and regular tetrahedra linked by triangles(SI Appendix, Fig. S18). Among these minimally frustrated 3-periodic hyperbolic triangulations with cubic symmetry, we havefound 10 infinite deltahedra, 6 of which have, to our knowl-edge, not been reported elsewhere. The remaining 4 were, toour knowledge, described in refs. 42 and 48. These deltahedraare shown in Fig. 4 and described in Table 2. All retain the

Table 2. Vertex-1–transitive cubic infinite deltahedra, 3N withequilateral triangular faces, shown in Fig. 4

Deltahedron NH2 NE3 Orbifold TPMS Symmetry Net

1 7 9 (×) 222321 D F4132 —2 (48) 7 7 (X) 23× (7) D Fd3 svu3 7 7 (X) 23× (8) D Fd3 svm4 7 7 (X) 23× (8) P I23 —5 8 7 (×) 222323 G1 I4132 tes6 (48) 8 8 (X) 222301 G2 I4132 nca7 (48) 8 8 (X) 2 ∗ 23 D Fd3m pyc8 (42) 9 9 (X) 2 ∗ 23 D Fd3m uty9 10 9 (×) 3 ∗ 22 P Pm3 shy10 12 9 (×) 2 ∗ 33 P P43m xay

“Embedded” polyhedra (marked with a X) have at most two facessharing any edge. G1 and G2 denote the two embeddings on the Gyroid.

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regular triangular facets of their parent hyperbolic triangu-lations, resulting in vertex-transitive cubic infinite deltahedra.However, these embeddings are not edge transitive, despitetheir equal lengths. The most symmetric examples are edge-2 transitive, and others are edge-3 and -4 transitive. Six ofthese infinite deltahedra are embedded, sponge-like polyhe-dra, with infinite 3-periodic internal channels. Some of thoseembedded deltahedra contain fragments of the regular Platonicdeltahedra: 2 is an array of face-sharing regular icosahedraand octahedra; 7 can be decomposed into face-sharing octa-hedra; and 4 and 6 have extended flat facets, containing mul-tiple triangular faces. Three of them (5, 9, and 10) collapseto lower-degree nets in E3, forming nonembedded deltahe-dra of types 38, 310, and 312. A fourth nonembedded examplecontains coincident vertices and edges (1). These four nonem-bedded examples contain fragments of the simpler convex Pla-tonic deltahedra: 1 comprises edge-shared regular octahedra andtetrahedra; 5, edge-shared octahedra; 9, edge-shared icosahedra;and 10, edge-shared tetrahedra.

Edge nets of some of these infinite polyhedra have beenreported previously (svu, nca, pyc, and xay in ref. 2) and are listedin RCSR (39). However, the deltahedra and simplicial polyhedraare structurally distinct from their skeletal nets.

ConclusionWe have described constructions in E3 of regular “hyperbolichoneycombs,” whose natural ambient space is H2. The minimallyfrustrated embeddings of hyperbolic honeycombs were built byminimal symmetry breaking of the honeycombs so that they can

be realized in E3. The five honeycombs analyzed here have min-imally frustrated embeddings on the cubic 3-periodic minimalsurfaces: the D , Gyroid , and P . Consequently, all of the patternsare realized in E3 with cubic symmetry. The 3-periodic crystallinestructures, including arrays of regular Platonic polyhedra andsphere packings, emerge therefore as least frustrated embed-dings in E3 of regular patterns in H2. The regular hyperbolichoneycombs map into E3 to give multiple minimally frustratedpatterns. We find 44 topologically distinct cubic nets, all with 2Dtopology [3, 7]; 24 [3, 8] nets; 12 [3, 9] nets; 3 [3, 10] nets; and3 [3, 12]nets. All of the minimally frustrated disc packings on theP and D surfaces adopt achiral symmetries, while their Gyroidanalogues are chiral. In contrast, the overwhelming majority ofthe nets and infinite polyhedra reported here are chiral, sincethey are induced by hyperbolic subgroups whose isometries donot include 2D reflections or glides. We found just 1, 5, 1, 2, and1 achiral [3, 7], [3, 8], [3, 9], [3, 10], and [3, 12]nets.

The construction pipeline outlined here can be generalized toarbitrary hyperbolic patterns. Regular examples, whose skeletalnets have topology {p, q} in H2, are the simplest generaliza-tions. Further, irregular hyperbolic z -valent nets with a rangeof ring topologies, such as examples with 2D vertex symbols(n1.n2 . . .nz ), can be embedded into E3 using this process.Such structures can require generalization of the definition ofa polyhedron, discussed in ref. 49.

ACKNOWLEDGMENTS. We thank Olaf Delgado-Friedrichs, Benedikt Kolbe,Stuart Ramsden, and Vanessa Robins for discussions related to this paper.M.C.P. acknowledges funding from the Carlsberg Foundation.

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